| Curve name |
$X_{193h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193h}$ minimally covers |
|
| Curves that minimally cover $X_{193h}$ |
|
| Curves that minimally cover $X_{193h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -28991029248t^{30} + 159450660864t^{28} - 338832654336t^{26} +
346080411648t^{24} - 196708663296t^{22} + 125306929152t^{20} -
102339182592t^{18} + 39537082368t^{16} - 6396198912t^{14} + 489480192t^{12} -
48024576t^{10} + 5280768t^{8} - 323136t^{6} + 9504t^{4} - 108t^{2}\]
\[B(t) = 1899956092796928t^{45} - 15674637765574656t^{43} +
54861232179511296t^{41} - 105863178545528832t^{39} + 118606243433545728t^{37} -
62247682575433728t^{35} - 24458223843016704t^{33} + 69236144121839616t^{31} -
52965392812867584t^{29} + 21023932022784000t^{27} - 5553079449550848t^{25} +
1388269862387712t^{23} - 328498937856000t^{21} + 51724016418816t^{19} -
4225838874624t^{17} + 93300719616t^{15} + 14841004032t^{13} - 1767370752t^{11} +
98592768t^{9} - 3193344t^{7} + 57024t^{5} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 30200268x - 37464430192$, with conductor $141120$ |
| Generic density of odd order reductions |
$139/1344$ |